A unified approach to combinatorial triangles: a generalized Eulerian polynomial

Abstract

Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array [T n,k ] n,k≥0 satisfying the recurrence relation:We derive a functional transformation for its row-generating function T n (x) from the row-generating function A n (x) of another array [A n,k ] n,k satisfying a two-term recurrence relation. Based on this transformation, we can get properties of T n,k and T n (x) including nonnegativity, log-concavity, real rootedness, explicit formula and so on. Then we extend the famous Frobenius formula, the γ positivity decomposition and the David-Barton formula for the classical Eulerian polynomial to those of a generalized Eulerian polynomial. We also get an identity for the generalized Eulerian polynomial with the general derivative polynomial. Finally, we apply our results to an array from the Lambert function, a triangular array from staircase tableaux and the alternating-runs triangle of type B in a unified approach.